Isometry between metric spaces is a very strong condition indeed. Often, a weaker condition is useful. For instance, any two norms on Rd(d<∞)induce quasi-isometric metric spaces; except when the two norms are multiples of each other, they never induce isometry.

If f is a quasi-isometry which is also onto, then the topologies of the two metric spaces are equivalent: convergence in the one implies convergence in the other.